Sławomir Czarnecki

The Isotropic and Cubic Material Designs. Recovery of the Underlying Microstructures Appearing in the Least Compliant Continuum Bodies

The paper discusses the problem of manufacturability of the minimum compliance designs of the structural elements made of two kinds of inhomogeneous materials: the isotropic and cubic. In both the cases the unit cost of the design is assumed as equal to the trace of the Hooke tensor. The Isotropic Material Design (IMD) delivers the optimal distribution of the bulk and shear moduli within the design domain. The Cubic Material Design (CMD) leads to the optimal material orientation and optimal distribution of the invariant moduli in the body made of the material of cubic symmetry. The present paper proves that the varying underlying microstructures (i.e., the representative volume elements (RVE) constructed of one or two isotropic materials) corresponding to the optimal designs constructed by IMD and CMD methods can be recovered by matching the values of the optimal moduli with the values of the effective moduli of the RVE computed by the theory of homogenization. The CMD method leads to a larger set of results, i.e., the set of pairs of optimal moduli. Moreover, special attention is focused on proper recovery of the microstructures in the auxetic sub-domains of the optimal designs.

The Free Material Design in Linear Elasticity

The Free Material Design (FMD) is a branch of topology optimization. In the present article the FMD formulation is confined to the minimum compliance problem within the linear elasticity setting. The design variables are all elastic moduli, forming a Hooke tensor C at each point of the design domain. The isoperimetric condition concerns the integral of the p-norm of the vector of the eigenvalues of the tensor C. The most important version refers to p = 1, imposing the condition on the integral of the trace of C. The paper delivers explicit stress-based formulations and numerical solutions of the FMD problems in the case of a single load case as well as for a general case of a finite number of load conditions.

Constrained Versions of the Free Material Design Methods and Their Applications in 3D Printing

The paper deals with the free material design and its constrained versions constructed by imposing: (a) cubic symmetry (cubic material design, CMD), (b) isotropy with: (b1) independent bulk and shear moduli (isotropic material design, IMD), and (b2) fixed Poisson’s ratio (Young’s modulus design, YMD). In the latter case the Young modulus is the only design variable. The moduli are viewed as non-negative, thus allowing for the appearance of void domains within the design domain. The paper shows that all these methods (CMD, IMD, YMD) reduce to two mutually dual problems: